Evidence

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verify/: The blind first review of the binary totient lead as it stood on 2026-10-03 and its distinct grade, which records pass and assigns no tier.

The entry point recomputes $v_2(\Phi_N)$ for $1\le N\le10^7$, where $\Phi_N=\sum_{n\le N}\varphi(n)2^{N-n}$, records the indices at which $v_2(\Phi_N)-\log_2N$ reaches a new maximum or exceeds $0,1,2,3$, checks that every record arises through a carry tie $v_2(\varphi(N))=1+v_2(\Phi_{N-1})$, counts the excess-positive indices $16\le N\le10^7$, those with $2^{v_2(\Phi_N)}>N+2$, lists the ones with $2^{v_2(\varphi(N))+2}>N+2$, counts the one-large-prime windows defined below, and certifies a prefix of the regular continued fraction of $S=\sum\varphi(n)2^{-n}$ from the bounds $\Phi_M/2^M\le S\le(\Phi_M+M+2)/2^M$ with $M=40000$. Run it from the repository root:

uv run --no-sync python wiki/research/leads/totient_binary_series/evidence/main.py

The default is the full declared domain; --quick uses $N\le10^6$ and $M=4000$ with its own fixed expectations. Both modes recompute everything from the totient and prime-factor sieves, read no files and write none, and exit nonzero on any failed obligation, including under python -O. The observed full run takes about ten seconds. The accumulator holds $\Phi_N$ modulo $2^{64}$, so a valuation is exact whenever it is below $64$; the run checks that the largest observed valuation, $29$, is below the cap. The failure protocol was checked by mutating one expected hit count and, after the extension, one expected window count in scratch copies; each produced a named failed obligation and exit code $1$.

The fixed expectations are the observed results stated on the owning page: the record list $(16,4),(28,6),(64,10),(88,12),(2146743,28),(2786632,29)$ of pairs $(N,v_2(\Phi_N))$; hit counts $72,45,21,9$ at excess thresholds $0,1,2,3$ over $16\le N\le10^7$; $71$ excess-positive indices in the same range, of which exactly $64,204,512,1632$ satisfy $2^{v_2(\varphi(N))+2}>N+2$; $240$ one-large-prime windows, the last at $N=31790$, with truncated-run statistic $20$ over $N\ge2^{15}-1$; a certified prefix of $11759$ partial quotients beginning $1;2,1,2,1,1,2,1,12$ with largest quotient $15531$ at index $7055$; and $6018$ decimal digits in the denominator of the last common convergent, which bounds the denominator of any rational value of $S$ because $S$ lies in the interval between the two bounds and hence in the cylinder of the common prefix. Record comparisons, threshold tests and the excess-positive and certificate conditions use integer comparisons; logarithms appear only in descriptive output. The empirical valuation distribution is printed as an observation and is not checked. This is author-checked finite coverage of a stated domain; it supplies no statement about all $N$, no standing for the target, and no independent review.

What the counts mean

The owning page proves that a dyadic rational value $S=a/2^{v}$ forces $2^{v_2(\Phi_N)}\le N+2$ once $N-v>\log_2(N+2)$, so each excess-positive index excludes the exponents $v<N-\log_2(N+2)$, and infinitely many would exclude all of them. The finite count $71$ says nothing about infinitely many. The page also proves that a residue class on which $2^m\mid\Phi_N$ holds for every sufficiently large member forces $2^{m-2}\mid\varphi(n)$ at every positive member $n$, so an excess-positive index that such a class could deliver must satisfy $2^{v_2(\varphi(N))+2}>N+2$. The list of the four indices meeting this necessary condition is a finite observation about the indices found, not a statement about the classes themselves. The [[research/leads/totient_binary_series/arithmetical_windows/classification|stronger classification]] proves separately that only $(n,m)=(5,3)$ can have actual divisibility, excess $2^m>n+2$, and an eventual whole-progression certificate together. Thus the four retained indices are candidates only for the coarser filter, and all fail the full class criterion. The checker retains that weaker diagnostic; its finite scan is not the proof of the classification.

An integer is called one-large-prime for a threshold $J$ when it has exactly one distinct prime factor above $J$. If $n=sq^a$ with $s$ $J$-smooth and $q>J$ prime, then $\varphi(n)=\varphi(s)q^{a-1}(q-1)$. The label describes a factorization pattern, not the general availability of exact factorization or of cancellation information. The window of $N$ is $N+1,\dots,N+J$ with $J=\lceil\log_2(N+2)\rceil$, a lower bound for the block length needed by the stated crude-tail certificate.

The script computes the largest and second-largest distinct prime factors of every integer up to $10^7+24$, controlled against $257$, $258=2\cdot3\cdot43$ and $264=2^3\cdot3\cdot11$. It reports the count and last index of full one-large-prime windows. Its additional run statistic is explicitly truncated: in the block $2^{J-1}-1\le N\le\min(2^J-2,L)$, it scans a run starting at $N+1$ only as far as $\min(2^J-2,L)+J$, where $L$ is the run's declared limit. The maximum of these truncated lengths for $N\ge2^{15}-1$ is $20$ when $L=10^7$. No claim about longer unscanned extensions is needed for the full-window test. The absence of full windows above $N=31790$ is a fact about this finite range only.

Why the denominator exclusion follows

Let $p/q$ be the convergent at the end of the common prefix and let $p'/q'$ be its predecessor. The closed cylinder has endpoints $p/q$ and $(p+p')/(q+q')$, in the order determined by parity, and $|pq'-p'q|=1$. The script checks that determinant and uses exact rational comparisons to place both tail bounds strictly between these endpoints. Every rational strictly inside the cylinder is $(pu+p'v)/(qu+q'v)$ for coprime integers $u>v\ge1$. The mediant endpoint has $u=v=1$; the convergent endpoint $p/q$ is treated separately. The determinant shows that the displayed numerator and denominator are coprime. Its denominator is therefore at least $q$; the convergent endpoint has denominator $q$ as well. This justifies the retained lower bound without inferring irrationality from a finite prefix.

Separate interpolation certificates

The [[research/leads/totient_binary_series/integer_interpolation/evidence/_index|integer interpolation evidence]] has its own entry point and retained exact inputs. It checks nine least full-image endpoints and all 36 specified central-zero target solutions, including their weighted full-tail costs. Its computation account gives the domain, commands and failure controls. The entry point on this page does not run those checks. Neither finite product establishes the uniform interpolation-cost estimate needed for irrationality.